We revisit theoretical background on OSIDH (Oriented Supersingular Isogeny Diffie-Hellman protocol), which is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. give a proof of fundamental theorem for OSIDH. The was stated without proof. Furthermore, we consider parameters OSIDH, sufficient condition the to work, estimate size certain security level.

We introduce the twisted μ4-normal form for elliptic curves, deriving in particular addition algorithms with complexity 9M+ 2S and doubling algorithms with complexity 2M + 5S + 2m over a binary field. Every ordinary elliptic curve over a finite field of characteristic 2 is isomorphic to one in this family. This improvement to the addition algorithm, applicable to a larger class of curves, is co...

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).